Anomalous vibrational properties in the continuum limit of glasses

The low-temperature thermal properties of glasses are anomalous with respect to those of crystals. These thermal anomalies indicate that the low-frequency vibrational properties of glasses differ from those of crystals. Recent studies revealed that, in the simplest model of glasses, i.e., the harmonic potential system, phonon modes coexist with soft localized modes in the low-frequency (continuum) limit. However, the nature of low-frequency vibrational modes of more realistic models is still controversial. In the present work, we study the Lennard-Jones (LJ) system using large-scale molecular-dynamics (MD) simulation and establish that the vibrational property of the LJ glass converges to coexistence of the phonon modes and the soft localized modes in the continuum limit as in the case of the harmonic potential system. Importantly, we find that the low-frequency vibrations are rather sensitive to the numerical scheme of potential truncation, which is usually implemented in the MD simulation, and this is the reason why contradictory arguments have been reported by previous works. We also discuss the physical origin of this sensitiveness by means of a linear stability analysis.

Figure 1

Vibrational modes in the LJ1 glass in (a) and the LJ0 glass in (b). The panels show the reduced vDOS $g\left(\omega \right)/{\omega }^2$ (top), the phonon order parameter $O^k$ (middle), and the participation ratio $P^k$ (bottom). In the top panel, we indicate the characteristic frequencies, ${\omega }_{\text{ex}0}$, ${\omega }_0$, ${\omega }_{\text{BP}}$, by arrows, and also show the Debye level $A_0$ by the horizontal line. The top panel in (a) shows $g\left(\omega \right)/{\omega }^2$ of extended modes ($P^k>{10}^{-2}$) and that of localized modes ($P^k<{10}^{-2}$). The inset to the middle panel in (a) plots the vDOS of localized modes ($P^k<{10}^{-2}$). In addition, the inset to the bottom panel in (a) shows the parametric plot of $O^k$ versus $P^k$ for different frequency regimes. Detailed discussions on these presented data are given in the main text.

Figure 2

Comparison of reduced vDOS between the LJ0 and LJ1 glasses. The data are the same as those presented in Fig. 1. For the harmonic potential system, the plateau is predicted around ${\omega }_{\text{BP}}$ by the mean-field theories [41, 44]. Although the present work studies the LJ potential system, not the harmonic potential system, the LJ1 glass seem to show plateau. In the LJ0 glass, the plateau is much narrower than in LJ1 glass or disappears.

Figure 3

Vibrational modes for several different cutoff distances $r_c$, for the LJ1 glass in (a) and (c) and the LJ0 glass in (b) and (d). The system size is $N=64000$. The panels (a) and (b) show the eigenvalue ${\lambda }^k$ as a function of the mode number $k$, for $r_c=2.5,3.0,3.5$, and 4.0. The panels (c) and (d) plot the participation ratio $P^k$ versus the eigenfrequency ${\omega }^k$, for $r_c=2.5$ and 4.0.

Figure 4

Minimum eigenvalue ${\lambda }_{\min }$ in the values of ${\lambda }^k$ that satisfy $O^k<0.4$, for the LJ0 glass. We plot ${\lambda }_{\min }/G\left(r_c\right)$ as a function of the cutoff length $r_c$ [$G\left(r_c\right)$ is the radial distirbution function at $r_c$]. The solid curve draws the theoretical prediction of ${\lambda }_{\min }/G\left(r_c\right)=32\pi \rho /r_c^5$ [see Eq. (11)]. The inset illustrates schematic description of our argument to derive Eq. (11).

Figure 5

Vibrational modes in the systems of 12IPL1 and 12IPL0. The system size is $N=64000$, and the cutoff distance is $r_c=1.8$. The panel (a) plots the eigenvalue ${\lambda }^k$ as a function of the mode number $k$. The panel (b) shows the participation ratio $P^k$ versus the eigenvalue ${\lambda }^k$ for the same eigenmodes as in (a).